ACARA v9 CONTENT DESCRIPTION “design and conduct repeated chance experiments and simulations, using digital tools to compare probabilities of simple events to related compound events, and describe results”
Builds on: relative frequency and combined events. This unit builds on listing outcomes and on relative frequency from data, turning them into designed experiments. Comparing simple and compound events through simulation completes the probability strand and the Year 9 course.
Estimating probability by experiment
Some probabilities are easy to work out by listing outcomes, but many real situations are too complex for that, or we simply want to check a theoretical answer against evidence. A simulation does exactly this: it models a chance situation using a simple random device, repeated many times, and uses the relative frequency of an outcome to estimate its probability. This unit is about designing and running such simulations, comparing simple events with related compound events, and describing what the results show.
Relative frequency and the law of large numbers
The link to relative frequency is the engine of the whole idea. When you repeat a chance experiment many times and record how often an event occurs, the relative frequency is an estimate of the true probability. The key fact, known informally as the law of large numbers, is that the more trials you run, the closer that estimate tends to get. Ten tosses of a coin might give six heads, a relative frequency of nought point six, but ten thousand tosses will usually land much nearer the true value of nought point five.
Relative frequency settles
As the number of tosses grows, the relative frequency of heads converges towards the theoretical 0.5.
Ten tosses might give 0.6, but ten thousand land near 0.5. More trials make the estimate converge on the theoretical probability.
Simulating with digital tools
This is where digital tools transform what is possible. By hand you might manage fifty trials; a random number generator or a spreadsheet can run ten thousand in an instant. To simulate a fair coin you generate a random zero or one; to simulate a fair die you generate a random whole number from one to six, each equally likely. The computer repeats the trial, tallies the outcomes, and reports the relative frequency, letting you see an estimate settle towards the theoretical probability before your eyes.
Simulating a coin and a die
A random 0 or 1 models a coin; a random whole number 1 to 6 models a fair die.
A random 0 or 1 simulates a coin; a random whole number 1 to 6 simulates a fair die. The tool tallies thousands of trials and reports the relative frequency.
Designing a simulation
Designing a good simulation follows a clear sequence. First identify the real event and the theoretical model behind it. Then choose a random device whose probabilities match: a coin for a one-half chance, a die for a one-sixth chance, or random numbers for anything in between. Define precisely what one trial is and what counts as a success. Repeat the trial many times using the digital tool, and finally use the relative frequency to estimate the probability and describe the result. A careless design, using a device with the wrong probabilities, would simulate the wrong situation entirely.
Designing a simulation
A five-step design cycle from identifying the event to estimating and describing the result.
Identify the event, choose a device whose probabilities match, define one trial, repeat many times with a digital tool, then estimate and describe the result.
Simple events versus compound events
The heart of this unit is comparing a simple event with a related compound event, because combining events can change a probability dramatically. The simple event of tossing one coin and getting a head has probability one half. Now consider the compound event of tossing three coins and getting at least one head. There are eight equally likely outcomes, and only one, three tails, has no head at all, so the probability of at least one head is one minus one eighth, which is seven eighths, or nought point eight seven five. The single head sits at one half, but the at-least-one head over three tosses is much higher.
Simple: one coin
One head from a single toss is one of two equally likely outcomes.
The simple event of one head on a single toss is one of two equally likely outcomes: P(head) = 1/2 = 0.5. This is the baseline.
Compound: at least one head in 3
Of the eight outcomes of three coins, only TTT has no head, so seven of eight succeed.
Three coins give 8 equally likely outcomes. Only TTT has no head, so P(at least one head) = 1 - 1/8 = 7/8 = 0.875.
Simple versus compound
The compound event of at least one head in three tosses is much more likely than one head on a single toss.
Side by side, the simple 0.5 and the compound 0.875 make the gap concrete: combining events raises the chance of at least one head.
A die example
A die gives the same lesson. The simple event of rolling one six has probability one sixth, about nought point one seven. The related compound event of rolling at least one six in four attempts is far more likely: the chance of no six in four rolls is five sixths multiplied by itself four times, leaving the probability of at least one six at about nought point five two. Simulating both, the simple event and the compound event, side by side makes the gap between them concrete rather than abstract.
Describing the results
Describing results is the final step, and it ties back to earlier ideas about evidence. A simulation gives an estimate, not an exact answer, so two runs may differ slightly, and a small number of trials gives a less reliable figure than a large one. A good description states the estimated probability, compares it with the theoretical value where one is known, and notes how many trials were used. Comparing the simple and compound estimates, and seeing both approach their theoretical values as trials increase, confirms that the simulation is behaving as it should. A simulation turns a chance question into an experiment: choose a matching random device, define a trial, repeat it many times with a digital tool, and read off the relative frequency as an estimate.
Quick self-check
1. A simulation estimates a probability using:
2. Why does running more trials in a simulation usually help?
3. To simulate a fair six-sided die with a digital tool, you generate:
4. Three coins are tossed. What is P(at least one head)?
5. Compare P(one head on a single toss) with P(at least one head in 3 tosses). Which is larger?